Addendum to “Termination of 4-fold Canonical Flips”

Addendum to “Termination of 4-fold Canonical Flips”

By

OsamuFujino^∗

Abstract

There are errors in the definition of the weighted version of ‘difficulty’ in “Ter- mination of 4-fold canonical flips”. In this paper, we describe these errors and correct them. After these corrections, Theorem 5.2 holds: every sequence of 4-fold canonical flips terminates.

§1. Introduction

Professor Alexeev pointed out that Lemma 2.1 in [F1], which is a copy of [K⁺, (4.12.2.1)], is wrong. Therefore, the weighted version of difficulty dS,b(X, B) in Definition 2.3 in [F1] is infinite ifb <maxj{bj}. So, the proof in [F1] is nonsense. In this paper, we change the definition ofdS,b(X, B) to make it finite when (X, B) is canonical and B has no reduced components, that is, the round down B = 0. Roughly speaking, in [F1, Definition 2.3] we ex- clude valuations obtained by one blow-up along generic points of codimension two subvarieties when we count valuations with small discrepancies. In this paper, we exclude valuations whose centers are codimension two subvarieties with good properties. By this change, the new version of dS,b(X, B) defined in Definition 4.4 becomes finite and the arguments in [F1] work without any changes. Proposition 3.1 is a key result in this paper. Note that the problems in [F1] are not in the arguments but in the definitions. As mentioned above,

Communicated by S. Mori. Received May 31, 2004.

2000 Mathematics Subject Classification(s): Primary 14E05; Secondary 14J35.

∗Graduate School of Mathematics, Nagoya University, Chikusa-ku Nagoya 464-8602, Japan.

e-mail: [email protected]

we have to assume B = 0 to make d_S,b(X, B) finite. Thus the main theo- rem: Theorem 1.1 in [F1] becomes slightly weaker. However, this assumption is harmless for applications if we use the special termination theorem (see [F3]).

For the precise statements of the termination theorems, see Theorems 5.1 and 5.2 below. Anyway, any sequence of4-fold canonical flips terminates.

We summarize the contents of this paper. In Section 2, we describe the errors in discrepancy lemmas in [K⁺, 4.12]. In Section 3, we formulate a new discrepancy lemma. Proposition 3.1 is the main result in this paper. In Section 4, we explain how to modify the definition of the weighted version of difficulty.

Section 5 is devoted to the statements of the termination of 4-fold canonical flips. We will use the same notation as in [F1] throughout this paper.

§2. Errors in Discrepancy Lemmas

The following example contradicts [F1, Lemma 2.1], which is a copy of [K⁺, (4.12.2.1)].

Example 2.1. LetX=P²,B= ²₃L, whereLis a line onX. LetP be any point onL. First, blow upX atP. Then we obtain an exceptional divisor E_P such that a(E_P, X, B) = ¹₃. Let L be the strict transform of L. Next, take a blow-up at L∩E_P. Then we obtain an exceptional divisor F_P whose discrepancya(FP, X, B) = ²₃. Note that thisFP isessential in the notation in [F1, Definition 2.1]. On the other hand, it is easy to see that discrep(X, B) =¹₃. Thus, min{1,1 + discrep(X, B)}= 1.

The proof of [K⁺, (4.12.2.1)] depends on [K⁺, (4.12.1.3)], which is obvi- ously wrong by Example 2.1 above. We need some extra assumption. It is not difficult to see that [K⁺, (4.12.1.3)] is true if we assume that bj ≤ ¹₂ for allj.

We write the precise statement for the reader’s convenience. This is essentially the same as [K, Corollary 3.2 (iii)] (see Remark 2.5 below).

Lemma 2.2. LetY be a smooth variety with a(not necessarily effective) Q-divisor B =

ibiBi such that

iBi has simple normal crossings, Bi is a prime divisor for every i,Bk =Bl fork=l, and thatbi≤ ¹₂ for alli. Assume that b_k+b_l≤0 wheneverB_k andB_l intersects. If ν is an algebraic valuation with small center on Y such that a(ν, Y, B)<1 thenν is obtained by blowing up the generic point of a subvarietyW ⊂Y such thatcodimYW = 2, only one of the Bk (say Bk₀)containsW andbk₀>0.

Remark2.3. In Example 2.1, we putD=dL. Thena(E_P, X, D) = 1−d.

Thus, the coefficient ofEP(resp.D, the strict transform ofD) isd−1 (resp.d).

Thus, (d−1) +d≤0 if and only if d≤ ¹₂. This computation shows that we have to assumeb_j≤ ¹₂ for allj in Lemma 2.2.

Thus we obtain the following lemma, which is a correction of [K⁺, (4.12.2.1)]. The proof is an exercise. Note that [K⁺, (4.12.2.2)] is contained in [KM, Proposition 2.36 (2)]. We do not needd_j ≤¹₂ for [K⁺, (4.12.2.2)].

Lemma 2.4. Let X be a normal variety andD=

jdjDj an effective Q-divisor onX such thatKX+DisQ-Cartier,whereDj is a prime divisor for every j and Dk =Dl fork=l. Assume that discrep(X, D)≥ −¹₂ anddj ≤¹₂ for all j. Let ν be an algebraic valuation with small center on X. Then there is a finite set of valuations {ν_i} such that if

a(ν, X, D)<min{1,1 + discrep(X, D)} and ν /∈ {νi}

thenνis obtained from blowing up the generic point of a subvarietyW ⊂D⊂X such that D andX are generically smooth along W (and thus only one of the D_j containsW)anddimW = dimX−2.

Unfortunately, Lemmas 2.2 and 2.4 are useless for our purpose. The as- sumption thatbj≤ ¹₂ for alljis too strong. Proposition 3.1 below seems to be a better formulation.

Remark2.5. Note that there are no problems in [K, Corollary 3.2 (iii)]

since Koll´ar assumed c > −¹₂ (for the notation, see Corollary 3.2 in [K]).

The assumption c > −¹₂ is in [K, Corollary 3.2 (ii)]. Lemma 2.2 in [M] is almost an exact copy of Corollary 3.2 in [K]. Therefore, [M, Lemma 2.2] is also correct. Matsuki gave me a comment about the remark which he made in [M, Lemma 2.2 (ii)] and which is not in [K, Corollary 3.2] “(actually >−1 is enough for the conclusion)”. This has to be understood that if we have the assumption 0≥c > −1, then the conclusion for (ii) holds (for the proof, see [KM, Proposition 2.36 (2)]), and NOT that the conclusion of (iii) holds (as Example 2.1 above is an obvious counter-example then). Thus, with the understanding that the assumptions are accumulative and not independent, it seems that the statements of the Corollary 3.2 in [K] and Lemma 2.2 in [M]

are correct and that the proof does not need any modifications. Therefore, the problems are not in [K] nor in [M], but in [K⁺, (4.12.1.3)]. For the finiteness of dN(X, D) in [K⁺, 4.12.3 Definition], we do not need [K⁺, (4.12.2.1)]. The statement [K⁺, (4.12.2.2)], which is true by [KM, Proposition 2.36 (2)], is sufficient. So, the error in [K⁺, (4.12.1.3)] causes no serious troubles in [K⁺, Chapter 4].

§3. New Discrepancy Lemma

The following proposition is a key result in this paper. The proof is es- sentially the same as one of [K⁺, (4.12.2.1)]. We give a proof for the reader’s convenience.

Proposition 3.1. Let X be a normal variety and B =

ib_iB_i a Q- divisor onX withB≤0,whereBiis a prime divisor for everyiandBk =Bl

for k=l. Assume thatKX+B is Q-Cartier and discrep(X, B)>−1. Note that (X, B) is called a sub klt pair in some literatures. Let ν be an algebraic valuation with small center onX. Then there is a finite set of valuations{νi} such that if

a(ν, X, D)<min{1,1 + discrep(X, D)} and ν /∈ {νi}

then V := CenterXν ⊂ B ⊂ X, B and X are generically smooth along V, dimV = dimX−2,and only one of the Bk (sayBk₀)containsV andbk₀>0.

Proof. First, we take a log resolutionf :Y −→X as in [KM, Proposition 2.36]. Thus, we have f^∗(KX+B) =KY +A−C, where A and C are both effective divisors with the following properties:

(i) A=

ai>0aiAiandC=

cj≥0cjCj have no common irreducible compo- nents,

(ii) Exc(f)∪Suppf_∗⁻¹B=

iA_i∪

jC_j, and (iii)

iA_i∪

jC_j is a simple normal crossing divisor and

iA_i is smooth.

Note that c_j may be zero and that A = f_∗⁻¹B+D, where D is an effective Q-divisor such that SuppD ∩ Suppf_∗⁻¹B = ∅. Next, if E is an exceptional divisor overY such thata(E, Y, A−C)<1, thenV := CenterYE⊂A⊂Y and dimV = dimY−2 by the following lemma: Lemma 3.2. We note that in general a(E, Y, F)≤a(E, Y, F) ifF≥F for any valuationE. IfV is contained inD, thena(E, Y, A−C)≥1 + discrep(X, B). Finally, the number of the exceptional divisors overY whose centers are inf_∗⁻¹B∩C witha(·, Y, A−C)<1 is finite (see Lemma 3.2 below), and it is obvious that the number of f-exceptional divisors is finite. Thus, we obtain the required finite set of valuations{νi}.

Lemma 3.2. Let Y be a smooth variety and H = dP, where P is a smooth prime divisor on Y and 0 < d < 1. Then discrep(Y, H) = 1−d.

If a(E, Y, H) < 1 for an exceptional divisor E over Y, then Center_YE is a codimension two subvariety ofY such thatCenterYE⊂P ⊂Y.

Let W be a codimension two subvariety of Y such that W ⊂ P ⊂ Y. Then there are only finitely many algebraic valuations ν’s with the following properties:

(1) a(ν, Y, H)<1, (2) Center_Yν=W.

Furthermore, ν attains the minimum,that is, a(ν, Y, H) = 1−d,if and only if ν is obtained by blowing upY along W.

Proof. This follows from easy computations. See [KM, Lemmas 2.45 and 2.29].

§4. How to Define a Weighted Difficulty

We introduce the notion ofsignificantdivisors. Proposition 3.1 and Lemma 2.2 imply that the notion of significant divisors are much better than one of essential divisors in [F1, Definition 2.3] for our purpose.

Definition 4.1. Let (X, B) be a canonical pair. We say that an ex- ceptional divisor E (over X) is significant unless W = CenterXE is a sub- variety W ⊂ B ⊂ X such that B and X are generically smooth along W (and thus only one of the irreducible components of SuppB contains W) and dimW = dimX−2.

The following corollary is obvious by Proposition 3.1. We will use this to define a weighted version of difficulty.

Corollary 4.2. Let (X, B)be a canonical pair withB= 0. Then we have

{E| E is significant and a(E, X, B)<1}<∞.

Remark4.3. Let (X, B) be a canonical pair. Assume thatB= 0. Let f :Y −→X be a log resolution of (X, B) withf^∗(KX+B) =KY +

iaiEi

such that

iEi = Exc(f)∪Suppf_∗⁻¹B. We can assume that a0 = 1. If E0

intersectsE₁such that 0≤a(E₁, X, B) =−a₁<1 and codim_Xf(E₀∩E₁)≥3, then we have infinitely many significant divisors whose centers aref(E₀∩E₁) witha(·, X, B) =−a1 by suitable blowing-ups whose centers are overE0∩E1. We define a weighted version of difficulty. To define this, we have to assume that the boundary divisor has no reduced components.

Definition 4.4 (A weighted version of difficulty). Let (X, B) be a pair with only canonical singularities, where B =l

j=1b_jB^j with 0 < b₁ <· · · <

bl<1 andB^jis a reduced divisor for everyj. We note thatB^jis not necessarily irreducible and that we assumebl<1. If (X, B) has only terminal singularities, thenB= 0. Thus the assumptionbl<1 always holds for terminal pairs. We put b₀= 0, andS:=

j≥0b_jZ_≥0⊂Q. Note that S= 0 ifB= 0. We set d_S,b(X, B) :=

ξ∈S,ξ≥b

{E|E is significant anda(E, X, B)<1−ξ}.

ThendS,b_j(X, B) is finite by Corollary 4.2.

§5. Statements of the Termination Theorems

Now the proof in [F1, §3] works without any changes only if we replace the word “essential” with “significant”. Thus we obtain the following theorem, which is slightly weaker than the original theorem: Theorem 1.1 in [F1].

Theorem 5.1. Let X be a normal projective4-fold and B an effective Q-divisor such that (X, B) is canonical andB= 0. Consider a sequence of log flips starting from(X, B) = (X0, B0):

(X0, B0) (X1, B1) (X2, B2) · · ·

Z₀ Z₁ ,

where φ_i:X_i−→Z_i is a contraction andφ_i⁺ :X_i⁺ =X_i+1 −→Z_i is the log flip. Then this sequence terminates after finitely many steps.

As we pointed out before,B = 0 if (X, B) has only terminal singulari- ties. Under the assumption that the varieties areQ-factorial and all the flipping contractions have the relative Picard number one, we obtain the following theo- rem by using the special termination theorem. These assumptions are harmless for applications.

Theorem 5.2. Let X be a normal projective4-fold and B an effective Q-divisor such that(X, B)is canonical. Assume that X is Q-factorial. Con- sider a sequence of log flips starting from(X, B) = (X₀, B₀):

(X0, B0) (X1, B1) (X2, B2) · · ·

Z₀ Z₁ ,

where φ_i:X_i−→Z_i is a contraction andφ_i⁺ :X_i⁺ =X_i+1−→Z_i is the log flip. We further assume that the relative Picard numberρ(X_i/Z_i) = 1for every i. Then this sequence terminates after finitely many steps.

Proof. By applying the special termination theorem (see [F3]) and shift- ing the index, we can assume that the flipping and flipped loci are disjoint fromBifor everyi. So, we can replaceBi with its fractional part. Thus this sequence terminates by Theorem 5.1.

Remark5.3. The final remark in [F1] should be removed. In [F2], we only need the termination of 4-fold semi-stable terminal flips. See Definition 2.3 in [F2]. Therefore, Theorems 5.1 and 5.2 are sufficient for [F2].

Acknowledgements

I would like to thank Professor Valery Alexeev, who pointed out an error in [F1], gave me comments, and obtained the same correction independently.

This paper was written in the Institute for Advanced Study. I am grateful to it for its hospitality. I was partially supported by a grant from the National Science Foundation: DMS-0111298. After I wrote the preliminary version of this paper, I received many useful comments from Professor Kenji Matsuki. I would like to thank him.

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